A radioactive substance has an initial mass of 160 grams. After 8 days, the remaining mass is 40 grams. Assuming...

1. TRANSLATE the problem information

• Given information:
  • Initial mass: 160 grams
  • Mass after 8 days: 40 grams
  • Formula: \(\mathrm{M = C(1/2)^{(t/r)}}\)
  • Need to find: \(\mathrm{r}\)

• What this tells us: \(\mathrm{C = 160}\) (initial mass), and we can use the 8-day data point to find \(\mathrm{r}\)

2. TRANSLATE the 8-day condition into an equation

• Substitute known values into \(\mathrm{M = C(1/2)^{(t/r)}}\):
  • \(\mathrm{M = 40, C = 160, t = 8}\)
  • \(\mathrm{40 = 160(1/2)^{(8/r)}}\)

3. SIMPLIFY to isolate the exponential term

• Divide both sides by 160:
  • \(\mathrm{40/160 = (1/2)^{(8/r)}}\)
  • \(\mathrm{1/4 = (1/2)^{(8/r)}}\)

4. INFER how to handle the exponential equation

• Key insight: Express \(\mathrm{1/4}\) as a power of \(\mathrm{1/2}\)
• Since \(\mathrm{1/4 = (1/2)^2}\), we have:
  • \(\mathrm{(1/2)^2 = (1/2)^{(8/r)}}\)

5. SIMPLIFY using exponent properties

• When bases are equal, exponents must be equal:
  • \(\mathrm{2 = 8/r}\)
• Solve for \(\mathrm{r}\):
  • \(\mathrm{r = 8/2 = 4}\)

Answer: C (4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that \(\mathrm{1/4}\) needs to be expressed as a power of \(\mathrm{1/2}\)

Many students get to \(\mathrm{1/4 = (1/2)^{(8/r)}}\) but then don't know how to proceed. They might try to convert to decimal form (\(\mathrm{0.25 = 0.5^{(8/r)}}\)) or attempt cross-multiplication incorrectly. Without recognizing that \(\mathrm{1/4 = (1/2)^2}\), they can't use the fundamental property that equal expressions with the same base have equal exponents.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors in the final step

After correctly setting up \(\mathrm{2 = 8/r}\), some students solve incorrectly, getting \(\mathrm{r = 2/8 = 1/4}\) instead of \(\mathrm{r = 8/2 = 4}\). This happens because they flip the fraction relationship.

This may lead them to select Choice B (1/4).

The Bottom Line:

This problem tests whether students can bridge the gap between recognizing an exponential equation setup and manipulating it using exponent properties. The key insight is expressing fractions as powers of the same base.